[1]} proposes 3 different training strategies for Concept Bottleneck Models (CBMs), we only consider joint and sequential model training strategy for all 3 datasets. As [1]} have demonstrated, joint model training mostly outperforms sequential training approach on task errors and has comparable performance on the concept errors. However, as already discussed previously, both sequential and joint model training have respective advantages with joint offering higher prediction accuracies while sequential offering more robust concept learning. In addition, joint model training is more time-efficient due to being end-to-end in nature and is more flexible to downstream tasks.
Although [1]} utilizes 3 different training objectives (Section 3, [1]}), we formulate them in a unified, general purpose objective function for all types of model training.
\(\mathcal {L}_{f,g} = \Sigma _{i}[\gamma *l(f(g(x_{i});y_{i}) + \lambda *\Sigma ^T_{j}l(g_j(x_{i});c^j_{i}) ]\) 
